Prime Packings of the Hexomino G: Torsten Sillke, FRA initial version: 1993 update: 20x21, 20x24 (Mike Reid) 1999-01 update: impossible 13-19 series (Mike Reid) 1999-01 update: impossible rectangles (Mike Reid) 1999-04-12 1 1 1 1 1 1 2-dim: ------ Zx 3p, 4p, 6, 7, 8 ... {6..8}+3n Nx 9, 12, 13, 14, 15, 16, 17, 18, 19, 20, ... {12..20}+9n 9x 12p, 20p, 24, 28p, ... {20,24,28}+12n 12x 9p, 13p, 14p, 17p, 18, 19p, 21p, 22, 23, 24p, 25p, 26, 27, 28, 29p ... {21..29}+9n 15x 28p, 32p, 36p, 40p, 44p, 48p, 52p, ... {28,32,36,40,44,48,52}+28n 18x 12, 16p, 20, ... {12, 16, 20}+12n 21x 12p, 20p, 24, 28, ... {20, 24, 28}+12n 24x 9, 12p, 13p, 14, 17, 18, 19, 20p, 21, 22, 23, 24, 25, ... {17..25}+9n 27x 12, 16p, 20, ... {12, 16, 20}+12n 30x 12, 16p, 20, ... {12, 16, 20}+12n 33x 12, 16p, 20, ... {12, 16, 20}+12n 36x 9, 12, 13, 14, 15p, 16, 17, 18, 19, 20, ... {12..20}+9n 39x 12, 16p, 20, ... {12, 16, 20}+12n 42x 12, 16p, 20, ... {12, 16, 20}+12n 45x 12, 16, 20, ... {12, 16, 20}+12n 48x 9, 12, 13, 14, 15p, 16, 17, 18, 19, 20, ... {12..20}+9n 3-dim: ------ 3x3x 8p, 12p, 16, 20, 24, 28 3x4x 5p, 6p, 7p, 9p, 10, 11, 12, 13, ... {9..13}+5n 3x5x 4p, 8, 10p, ... {8, 10}+4n 3x6x 4p, 6p, 7p, 8, 9p, ... {6..9}+4n Impossible: Zx5 Nx{3,4,5,6,7,8,10,11} n x m with n and m are no multiples of 4 (Mike Reid) 9x16 12x{12,15,16,20} 15x{16,20,24} 21x{16} 24x{15,16} 3x3x{4,6,10,14,18,22,26,30} 3x4x{3,4,8} 3x5x6 3x6x{3,5} Annotations: 3xZ strip: A A A B A A B B ... A B B B 4xZ strip: B B B B B A ... B A A A A A 9xN strips: These dissect into 9x(12+8n) rectangles. Therefore no odd 9xk rectangle is possible. 3 3 2 2 2 1 1 1 2 2 3 3 3 x x x x x x x 3 3 3 2 2 1 1 1 2 2 2 3 3 2 x x x x x x 3 1 1 2 1 1 1 3 2 1 1 3 2 2 x x x x x x 1 1 1 3 1 1 1 3 3 1 1 1 2 2 2 x x x x x 2 2 1 3 3 3 ... 3 3 3 1 2 3 3 3 ... x x x x x x 2 2 2 3 3 1 1 1 2 2 2 3 3 x x x x x x x 2 3 1 1 1 2 1 1 3 2 2 1 3 2 x x x x x x 3 3 1 1 2 2 1 3 3 1 1 1 2 2 x x x x x x 3 3 3 1 2 2 2 3 3 3 1 1 2 2 2 x x x x x the 15x(28+4n) rectangles: (15x28 has 2 solutions) 84 84 80 80 80 75 68 68 68 62 62 57 57 57 49 49 45 45 45 38 38 32 32 32 A A A A 84 84 84 80 80 75 75 68 68 62 62 62 57 57 49 49 49 45 45 38 38 38 32 32 A A A A 84 85 85 80 75 75 75 68 69 62 63 63 57 55 49 50 50 45 43 38 39 39 32 A A A A 21 85 85 85 78 78 78 69 69 69 63 63 63 55 55 50 50 50 43 43 39 39 39 30 30 30 21 21 21 86 86 85 78 78 73 73 69 69 64 64 63 55 55 55 51 50 43 43 43 35 39 30 30 26 26 21 21 86 86 86 82 78 73 73 73 70 64 64 64 56 56 56 51 51 44 44 44 35 35 35 30 26 26 26 22 86 82 82 82 76 73 70 70 70 64 61 61 56 56 51 51 51 44 44 40 35 35 33 33 26 22 22 22 87 87 82 82 76 76 76 70 70 66 61 61 61 56 53 48 48 48 44 40 40 33 33 33 27 27 22 22 87 87 87 83 76 76 74 74 66 66 61 58 58 58 53 53 48 48 40 40 40 34 34 33 27 27 27 23 87 83 83 83 79 74 74 74 66 66 66 58 58 53 53 53 48 B B B B 34 34 34 27 23 23 23 88 88 83 83 79 79 71 74 67 67 67 59 58 54 54 54 B B B B 36 34 31 31 31 24 23 23 88 88 88 79 79 79 71 71 71 67 67 59 59 59 54 54 B B B B 36 36 36 31 31 24 24 24 88 89 81 81 81 77 71 71 72 67 65 59 59 60 54 52 A A A A 36 36 37 31 29 24 24 25 89 89 81 81 77 77 72 72 72 65 65 60 60 60 52 52 A A A A 37 37 37 29 29 25 25 25 89 89 89 81 77 77 77 72 72 65 65 65 60 60 52 52 52 A A A A 37 37 29 29 29 25 25 this can be extended by: A A A A A A A A A A A A B B B B B B B B B B B B A A A A A A A A A A A A B B B B B B B B B B B B A A A A A A A A A A A A unique packings: 9x{12,20,28} 12x{13,14,17,19,21,24,25} (12x29 has two solutions) The 16x3k series is complete. the 3x5x10 box (an odd box): 89 84 82 82 82 77 72 72 72 66 88 83 83 82 82 76 71 72 72 66 88 88 88 82 78 78 71 72 67 66 88 88 80 80 80 75 71 67 67 65 85 85 85 79 74 74 74 67 67 67 89 84 84 84 77 77 73 73 73 66 89 83 83 83 76 76 71 73 73 66 89 86 86 78 78 78 71 73 68 65 86 86 86 80 80 75 75 68 68 65 85 85 86 79 79 74 74 68 68 68 89 84 84 81 77 77 77 69 69 69 89 83 81 81 76 76 76 69 69 66 87 87 81 81 81 78 71 70 69 65 87 87 87 80 75 75 75 70 70 65 87 85 79 79 79 74 70 70 70 65 the 3x6x6 box (unique): 89 85 81 81 81 73 88 88 82 79 75 78 88 88 88 78 78 78 88 84 80 76 78 78 86 84 80 76 74 72 86 84 83 76 74 72 89 85 82 81 81 73 89 85 82 79 75 73 89 85 80 79 75 73 86 84 80 79 74 72 86 84 80 76 74 72 86 83 83 76 74 72 89 85 82 81 75 73 89 85 82 79 75 73 87 87 82 79 75 77 87 87 87 77 77 77 87 84 80 76 77 77 86 83 83 83 74 72 Free Group Technique: (A) (B) x x x x x and x x have the same path group. x x x x x (B) -> (A) 1 2 2 2 1 1 1 2 1 2 1 2 = 1 2 2 2 1 1 1 2 1 1 2 2 (A) -> (B) 1 1 1 x x x 1 1 x x x 1 x x x x x x -> 1 1 1 x x x 1 1 - x x 1 x - - - - - -> 1 1 1 x x x 1 1 y x x 1 x y y y y y -> 1 1 1 x x x 1 1 y - x 1 - - - y - - -> - - - x x x - - y x - y The tiling group of x x x x x x x and x x contains x x x x x x x x x AS: b c a a b b - c c c a b b b = - c - c a a a - - - References: - Ross Honsberger; Mathematical Gems II, MAA 1976, The Dolciani Mathematical Expositions 2. (german: Mathematische Juwelen, Vieweg, 1982) Chapter 4: Packing Problems - David. A. Klarner; Packing a rectangle with congruent n-ominoes, Journal of Combinatorial Theory 7 (1969) 107-115 - David A. Klarner, Frits G"obel; Packing boxes with congruent figures, Indag. Math. 31 (1969) 465-472 - Michael Reid; one side length must by a multiple of 4. email from 1999-04-12